Introduction - KTHdary/luna-field20190318.pdf · A LUNA ETALE SLICE THEOREM FOR ALGEBRAIC STACKS JAROD ALPER, JACK HALL, AND DAVID RYDH Abstract. We prove that every algebraic stack,

A LUNA ETALE SLICE THEOREM FOR ALGEBRAIC STACKS

JAROD ALPER, JACK HALL, AND DAVID RYDH

Abstract. We prove that every algebraic stack, locally of finite type over an

algebraically closed field with affine stabilizers, is etale-locally a quotient stackin a neighborhood of a point with a linearly reductive stabilizer group. The

proof uses an equivariant version of Artin’s algebraization theorem proved in

the appendix. We provide numerous applications of the main theorems.

1. Introduction

Quotient stacks form a distinguished class of algebraic stacks which provide in-tuition for the geometry of general algebraic stacks. Indeed, equivariant algebraicgeometry has a long history with a wealth of tools at its disposal. Thus, it haslong been desired—and more recently believed [Alp10, AK16]—that certain alge-braic stacks are locally quotient stacks. This is fulfilled by the main result of thispaper: at a point with linearly reductive stabilizer, an algebraic stack is etale-locally a quotient stack of an affine scheme by the stabilizer. In the case of smoothalgebraic stacks, we can provide a more refined description which resolves thealgebro-geometric counterpart to the Weinstein conjectures [Wei00]—now knownas Zung’s Theorem [Zun06, CF11, CS13, PPT14]—on the linearization of properLie groupoids in differential geometry.

What the main theorems (Theorem 1.1 and 1.2) of this paper really justifyis a philosophy that quotient stacks of the form [SpecA/G], where G is a linearlyreductive group, are the building blocks of algebraic stacks near points with linearlyreductive stabilizers. These theorems yield a number of applications to old and newproblems, including:

• a generalization of Luna’s etale slice theorem to non-affine schemes (§2.2);• a generalization of Sumihiro’s theorem on torus actions to Deligne–Mumford

stacks (§2.3), confirming an expectation of Oprea [Opr06, §2];• Bia lynicki-Birula decompositions for smooth Deligne–Mumford stacks (§2.4),

generalizing Skowera [Sko13];• a criterion for the existence of a good moduli space (§2.9), generalizing

[KM97, AFS17];• a criterion for etale-local equivalence of algebraic stacks (§2.8), extending

Artin’s corresponding results for schemes [Art69a, Cor. 2.6];• the existence of equivariant miniversal deformation spaces for curves (§2.5),

generalizing [AK16];• a characterization of toric Artin stacks in terms of stacky fans [GS15,

Thm. 6.1] (§2.12);• the etale-local quotient structure of a good moduli space (§2.6);

Date: Mar 18, 2019.

2010 Mathematics Subject Classification. Primary 14D23; Secondary 14B12, 14L24, 14L30.During the preparation of this paper, the first author was partially supported by the Australian

Research Council grant DE140101519 and by a Humboldt Fellowship. The second author was

partially supported by the Australian Research Council. The third author is supported by theSwedish Research Council 2011-5599.

1

2 J. ALPER, J. HALL, AND D. RYDH

• formal GAGA for good moduli space morphisms (§2.6), resolving a conjec-ture of Geraschenko–Zureick-Brown [GZB15, Conj. 32];• a short proof of Drinfeld’s results [Dri13] on algebraic spaces with a Gm-

action (§2.11);• compact generation of derived categories of algebraic stacks (§2.13); and• a resolution of a wall-crossing conjecture for higher rank DT/PT invariants

by Toda [Tod16].

Our first theorem gives a precise description of the etale-local structure of analgebraic stack at a non-singular point with linearly reductive stabilizer. This isthe algebro-geometric analogue of Zung’s resolution of the Weinstein conjectures.

Before we state the theorem, we introduce the following notation: if X is an alge-braic stack over a field k and x ∈ X(k) is a closed point with stabilizer group schemeGx, then we let Nx denote the normal space to x viewed as a Gx-representation. IfI ⊆ OX denotes the sheaf of ideals defining x, then Nx = (I/I2)∨. If Gx is smooth,then Nx is identified with the tangent space of X at x; see Section 4.1.

Theorem 1.1. Let X be a quasi-separated algebraic stack, locally of finite type overan algebraically closed field k, with affine stabilizers. Let x ∈ |X| be a smooth andclosed point with linearly reductive stabilizer group Gx. Then there exists an affineand etale morphism (U, u)→ (Nx//Gx, 0), where Nx//Gx denotes the GIT quotient,and a cartesian diagram

([Nx/Gx], 0

)

([W/Gx], w

) f //

oo (X, x)

(Nx//Gx, 0) (U, u)oo

such that W is affine and f is etale and induces an isomorphism of stabilizer groupsat w. In addition, if X has affine diagonal, the morphism f can be arranged to beaffine.

In particular, this theorem implies that X and [Nx/Gx] have a common etaleneighborhood of the form [SpecA/Gx]. Our second theorem gives a local descriptionof an algebraic stack at a (potentially singular) point with respect to a linearlyreductive subgroup of the stabilizer.

Theorem 1.2. Let X be a quasi-separated algebraic stack, locally of finite type overan algebraically closed field k, with affine stabilizers. Let x ∈ X(k) be a point andH ⊆ Gx be a subgroup scheme of the stabilizer such that H is linearly reductive andGx/H is smooth (resp. etale). Then there exists an affine scheme SpecA with anaction of H, a k-point w ∈ SpecA fixed by H, and a smooth (resp. etale) morphism

f :([SpecA/H], w

)→ (X, x)

such that BH ∼= f−1(BGx); in particular, f induces the given inclusion H → Gxon stabilizer group schemes at w. In addition, if X has affine diagonal, then themorphism f can be arranged to be affine.

The main techniques employed in the proof of Theorem 1.1 are

(1) deformation theory,(2) coherent completeness,(3) Tannaka duality, and(4) Artin approximation.

A LUNA ETALE SLICE THEOREM FOR ALGEBRAIC STACKS 3

Deformation theory produces an isomorphism between the nth infinitesimal neigh-

borhood N[n]x of 0 in Nx = [Nx/Gx] and X[n], the nth infinitesimal neighbor-

hood of x in X. It is not at all obvious, however, that the system of morphisms

f [n] : N[n]x → X algebraizes. We establish algebraization in two steps.

The first step is effectivization. To accomplish this, we introduce coherent com-pleteness, a key concept of the article. Recall that if (A,m) is a complete local ring,then Coh(A) = lim←−n Coh(A/mn+1). Coherent completeness is a generalization of

this, which is more refined than the formal GAGA results of [EGA, III.5.1.4] and[GZB15] (see §2.6). What we prove in §3.1 is the following.

Theorem 1.3. Let G be a linearly reductive affine group scheme over an alge-braically closed field k. Let SpecA be a noetherian affine scheme with an actionof G, and let x ∈ SpecA be a k-point fixed by G. Suppose that AG is a completelocal ring. Let X = [SpecA/G] and let X[n] be the nth infinitesimal neighborhood ofx. Then the natural functor

(1.1) Coh(X)→ lim←−n

Coh(X[n]

)is an equivalence of categories.

Tannaka duality for algebraic stacks with affine stabilizers was recently estab-lished by the second two authors [HR19, Thm. 1.1] (also see Theorem 3.5). Thisproves that morphisms between algebraic stacks Y → X are equivalent to sym-metric monoidal functors Coh(X) → Coh(Y). Therefore, to prove Theorem 1.1,we can combine Theorem 1.3 with Tannaka duality (Corollary 3.6) and the above

deformation-theoretic observations to show that the morphisms f [n] : N[n]x → X

effectivize to f : Nx → X, where Nx = Nx ×Nx//GxSpec ONX//Gx,0. The morphism

f is then algebraized using Artin approximation [Art69a].The techniques employed in the proof of Theorem 1.2 are similar, but the meth-

ods are more involved. Since we no longer assume that x ∈ X(k) is a non-singularpoint, we cannot expect an etale or smooth morphism Nx → X. Using Theorem

1.3 and Tannaka duality, however, we can produce a closed substack H of Nx and

a formally versal morphism f : H → X. To algebraize f , we apply an equivariantversion of Artin algebraization (Corollary A.16), which we believe is of independentinterest.

For tame stacks with finite inertia, Theorems 1.1 and 1.2 are one of the mainresults of [AOV08]. The structure of algebraic stacks with infinite stabilizers hasbeen poorly understood until the present article. For algebraic stacks with infinitestabilizers that are not—or are not known to be—quotient stacks, Theorems 1.1and 1.2 were only known when X = Mss

g,n is the moduli stack of semistable curves.This is the central result of [AK16], where it is also shown that f can be arranged tobe representable. For certain quotient stacks, Theorems 1.1 and 1.2 can be obtainedusing traditional methods in equivariant algebraic geometry, see §2.2 for details.

1.1. Some remarks on the hypotheses. We mention here several examples il-lustrating the necessity of some of the hypotheses of Theorems 1.1 and 1.2.

Example 1.4. Some reductivity assumption of the stabilizer Gx is necessary inTheorem 1.2. For instance, consider the group scheme G = Spec k[x, y]xy+1 →A1 = Spec k[x] (with multiplication defined by y 7→ xyy′ + y + y′), where thegeneric fiber is Gm but the fiber over the origin is Ga. Let X = BG and x ∈ |X|be the point corresponding to the origin. There does not exist an etale morphism([W/Ga], w)→ (X, x), where W is an algebraic space over k with an action of Ga.


Example 1.5. It is essential to require that the stabilizer groups are affine in aneighborhood of x ∈ |X|. For instance, let X be a smooth curve and E → X bea group scheme whose generic fiber is a smooth elliptic curve but the fiber overa point x ∈ X is isomorphic to Gm. Let X = BE. There is no etale morphism([W/Gm], w)→ (X, x), where W is an affine k-scheme with an action of Gm.

Example 1.6. In the context of Theorem 1.2, it is not possible in general to finda Zariski-local quotient presentation of the form [SpecA/Gx]. Indeed, if C is theprojective nodal cubic curve with Gm-action, then there is no Zariski-open Gm-invariant affine neighborhood of the node. If we view C (Gm-equivariantly) as theZ/2-quotient of the union of two P1’s glued along two nodes, then after removingone of the nodes, we obtain a (non-finite) etale morphism [Spec(k[x, y]/xy)/Gm]→[C/Gm] where x and y have weights 1 and −1. This is in fact the unique suchquotient presentation (see Remark 2.13).

Example 1.7. Let Log (resp. Logal) be the algebraic stack of log structures (resp.aligned log structures) over Spec k introduced in [Ols03] (resp. [ACFW13]). Let Nrbe the free log structure on Spec k. There is an etale neighborhood [Spec k[Nr]/GrmoSr] → Log of Nr which is not representable. Note that Log does not have sepa-rated diagonal. Similarly, there is an etale neighborhood [Spec k[Nr]/Grm]→ Logal

of Nr (with the standard alignment) which is representable but not separated. Be-cause [Spec k[Nr]/Grm] → Logal is inertia-preserving, Logal has affine inertia andhence separated diagonal; however, the diagonal is not affine. In both cases, thisis the unique such quotient presentation (see Remark 2.13). These examples illus-trate that it is not always possible to obtain a representable or separated quotientpresentation in Theorem 1.2 without additional hypotheses; see also Question 1.9.

1.2. Generalizations. Using a similar argument, one can in fact establish a gen-eralization of Theorem 1.2 to the relative and mixed characteristic setting. Thisrequires developing some background material on deformations of linearly reduc-tive group schemes, a more general version of Theorem 1.3 and a generalization ofthe formal functions theorem for good moduli spaces. To make this paper moreaccessible, we have decided to postpone the relative statement until a future paper.

If Gx is not reductive, it is possible that one could find an etale neighborhood([SpecA/GLn], w)→ (X, x). However, this is not known even if X = Bk[ε]Gε whereGε is a deformation of a non-reductive algebraic group [Con10].

In characteristic p, the linearly reductive hypothesis in Theorem 1.2 is quiterestrictive. Indeed, an algebraic group G over an algebraically closed field k ofcharacteristic p is linearly reductive if and only if G0 is a torus and |G/G0| iscoprime to p [Nag62]. We ask however:

Question 1.8. Does a variant of Theorems 1.1 and 1.2 remain true if “linearlyreductive” is replaced with “reductive”?

We remark that if X is a Deligne–Mumford stack, then the conclusion of Theorem1.2 holds. We also ask:

Question 1.9. If X has separated (resp. quasi-affine) diagonal, then can the mor-phism f in Theorems 1.1 and 1.2 be chosen to be representable (resp. quasi-affine)?

If X does not have separated diagonal, then the morphism f cannot necessarilybe chosen to be representable. For instance, consider the non-separated affine lineas a group scheme G → A1 whose generic fiber is trivial but the fiber over theorigin is Z2. Then BG admits an etale neighborhood f : [A1/Z2] → BG whichinduces an isomorphism of stabilizer groups at 0, but f is not representable in aneighborhood. Example 1.7 provides another example. We answer Question 1.9


affirmatively when X has affine diagonal, or is a quotient stack (Proposition 4.2),or when H is diagonalizable (Proposition 4.3).

1.3. Notation. An algebraic stack X is quasi-separated if the diagonal and thediagonal of the diagonal are quasi-compact. An algebraic stack X has affine stabi-lizers if for every field k and point x : Spec k → X, the stabilizer group Gx is affine.

If X is an algebraic stack and Z ⊆ X is a closed substack, we will denote by X[n]Z

the nth nilpotent thickening of Z ⊆ X (i.e., if I ⊆ OX is the ideal sheaf defining Z,

then X[n]Z → X is defined by In+1). If x ∈ |X| is a closed point, the nth infinitesimal

neighborhood of x is the nth nilpotent thickening of the inclusion of the residualgerbe Gx → X.

Recall from [Alp13] that a quasi-separated and quasi-compact morphism φ : X→Y of algebraic stacks is cohomologically affine if the push-forward functor φ∗ onthe category of quasi-coherent OX-modules is exact. If Y has quasi-affine diag-onal and φ has affine diagonal, then φ is cohomologically affine if and only ifRφ∗ : D+

QCoh(X) → D+QCoh(Y) is t-exact, cf. [Alp13, Prop. 3.10 (vii)] and [HNR18,

Prop. 2.1]; this equivalence is false if Y does not have affine stabilizers [HR15,Rem. 1.6]. If G → Spec k is an affine group scheme of finite type, we say that Gis linearly reductive if BG → Spec k is cohomologically affine. We say that G isan algebraic group over k if G is a smooth, affine group scheme over k. A quasi-separated and quasi-compact morphism φ : X → Y of algebraic stacks is a goodmoduli space if Y is an algebraic space, φ is cohomologically affine and OY → φ∗OX

is an isomorphism.If G is an affine group scheme of finite type over k acting on an algebraic space

X, we say that a G-invariant morphism π : X → Y of algebraic spaces is a good GITquotient if the induced map [X/G] → Y is a good moduli space; we often writeY = X//G. In the case that G is linearly reductive, a G-equivariant morphismπ : X → Y is a good GIT quotient if and only if π is affine and OY → (π∗OX)G isan isomorphism.

If X is a noetherian algebraic stack, we denote by Coh(X) the category of coherentOX-modules.

Acknowledgements. We thank Andrew Kresch for many useful conversations.We also thank Bjorn Poonen for suggesting the argument of Lemma 5.1. We wouldalso like to thank Dragos Oprea and Michel Brion for some helpful comments.

2. Applications

Theorems 1.1 and 1.2 have many striking applications. We first record someimmediate consequences of Theorems 1.1 and 1.2. Unless stated otherwise, for thissection k will denote an algebraically closed field.

2.1. Immediate consequences. If X is a quasi-separated algebraic stack, locallyof finite type over k with affine stabilizers, and x ∈ X(k) has linearly reductivestabilizer Gx, then

(1) there is an etale neighborhood of x with a closed embedding into a smoothalgebraic stack;

(2) there is an etale-local description of the cotangent complex LX/k of X interms of the cotangent complex LW/k of W = [SpecA/Gx]. If x ∈ |X| is asmooth point (so that W can be taken to be smooth), then LW/k admitsan explicit description. In general, the [0, 1]-truncation of LW/k can bedescribed explicitly by appealing to (1);

(3) for any representation V of Gx, there exists a vector bundle over an etaleneighborhood of x extending V ;


(4) the Gx-invariants of a formal miniversal deformation space of x is isomor-phic to the completion of a finite type k-algebra.

(5) any specialization y x of k-points is realized by a morphism [A1/Gm]→X. This follows by applying the Hilbert–Mumford criterion to an etalequotient presentation constructed by Theorem 1.2.

We now state some further applications of Theorems 1.1 and 1.2. We will defertheir proofs until §5.

2.2. Generalization of Luna’s etale slice theorem. We now provide a refine-ment of Theorem 1.2 in the case that X = [X/G] is a quotient stack. The followingtheorem provides a generalization of Luna’s etale slice theorem.

Theorem 2.1. Let X be a quasi-separated algebraic space, locally of finite typeover k, with an action of an algebraic group G. Let x ∈ X(k) be a point witha linearly reductive stabilizer Gx. Then there exists an affine scheme W with anaction of Gx which fixes a point w, and an unramified Gx-equivariant morphism

(W,w)→ (X,x) such that f : W ×Gx G→ X is etale.1

If X admits a good GIT quotient X → X//G, then it is possible to arrange thatthe induced morphism W//Gx → X//G is etale and W ×Gx G ∼= W//Gx ×X//G X.

Let Nx = TX,x/TG·x,x be the normal space to the orbit at x; this inherits anatural linear action of Gx. If x ∈ X is smooth, then it can be arranged that thereis an etale Gx-equivariant W → Nx such that W//Gx → Nx//Gx is etale and

Nx ×Gx G

W ×Gx Gf //

oo X

Nx//Gx W//Gxoo

is cartesian.

Remark 2.2. The theorem above follows from Luna’s etale slice theorem [Lun73] ifX is affine. In this case, Luna’s etale slice theorem is stronger than Theorem 2.1 asit asserts additionally that W → X can be arranged to be a locally closed immersion(which is obtained by choosing a Gx-equivariant section of TX,x → Nx and thenrestricting to an open subscheme of the inverse image of Nx under a Gx-equivariantetale morphism X → TX,x). Note that while [Lun73] assumes that char(k) = 0 andG is reductive, the argument goes through unchanged in arbitrary characteristicif G is smooth, and Gx is smooth and linearly reductive. Moreover, with minormodifications, the argument in [Lun73] is also valid if Gx is not necessarily smooth.

Remark 2.3. More generally, if X is a normal scheme, it is shown in [AK16, §2.1]that W → X can be arranged to be a locally closed immersion. However, when Xis not normal or is not a scheme, one cannot always arrange W → X to be a locallyclosed immersion and therefore we must allow unramified “slices” in the theoremabove.

2.3. Generalization of Sumihiro’s theorem on torus actions. In [Opr06,§2], Oprea speculates that every quasi-compact Deligne–Mumford stack X witha torus action has an equivariant etale atlas SpecA → X. He proves this whenX = M0,n(Pr, d) is the moduli space of stable maps and the action is induced byany action of Gm on Pr and obtains some nice applications. We show that Oprea’sspeculation holds in general.

1Here, W ×Gx G denotes the quotient (W × G)/Gx. Note that there is an identification ofGIT quotients (W ×Gx G)//G ∼= W//Gx.


Let T be a torus acting on an algebraic stack X, locally of finite type over k,via σ : T × X → X. Let Y = [X/T ]. Let x ∈ X(k) be a point with image y ∈ Y(k).There is an exact sequence

(2.1) 1→ Gx → Gy → Tx → 1

where the stabilizer Tx ⊆ T is defined by the fiber product

Tx ×BGxσx //

BGx

T ×BGx

σ|x //

X

(2.2)

and σ|x : T × BGx(id,ιx)−−−−→ T × X

σ−→ X. Observe that Gy = Spec k ×BGxTx. The

exact sequence (2.1) is trivially split if and only if the induced action σx of Tx onBGx is trivial. The sequence is split if and only if the action σx comes from a grouphomomorphism T → Aut(Gx).

Theorem 2.4. Let X be a quasi-separated algebraic (resp. Deligne–Mumford) stack,locally of finite type over k, with affine stabilizers. Let T be a torus with an actionon X. Let x ∈ X(k) be a point such that Gx is smooth and the exact sequence (2.1)is split (e.g., X is an algebraic space). There exists a T -equivariant smooth (resp.etale) neighborhood (SpecA, u)→ (X, x) that induces an isomorphism of stabilizersat u.

The theorem above fails when (2.1) does not split. For a simple example, considerthe action of T = Gm on X = Bµµµn defined by: for t ∈ Gm(S) = Γ(S,OS)∗

and (L, α) ∈ Bµµµn(S) (where L is a line bundle on S and α : L⊗n → OS is anisomorphism), then t · (L, α) = (L, t α). The exact sequence of (2.1) is 1 →µµµn → Gm

n−→ Gm → 1 which does not split. In this case though, there is an etale

presentation Spec k → Bµµµn which is equivariant under Gmn−→ Gm. More generally,

we have:

Theorem 2.5. Let X be a quasi-separated Deligne–Mumford stack, locally of finitetype over k. Let T be a torus with an action on X. If x ∈ X(k), then there exist areparameterization α : T → T and an etale neighborhood (SpecA, u) → (X, x) thatis equivariant with respect to α.

In the case that X is a normal scheme, Theorem 2.4 was proved by Sumi-hiro [Sum74, Cor. 2], [Sum75, Cor. 3.11]; then SpecA → X can be taken to bean open neighborhood. The nodal cubic with a Gm-action provides an examplewhere there does not exist a Gm-invariant affine open cover. Theorem 2.4 was alsoknown if X is a quasi-projective scheme [Bri15, Thm. 1.1(iii)] or if X is a smooth,proper, tame and irreducible Deligne–Mumford stack, whose generic stabilizer istrivial and whose coarse moduli space is a scheme [Sko13, Prop. 3.2]. We can alsoprove:

Theorem 2.6. Let X be a quasi-separated algebraic space, locally of finite type overk, with an action of an affine group scheme G of finite type over k. Let x ∈ X(k)be a point with linearly reductive stabilizer Gx. Then there exists an affine schemeW with an action of G and a G-equivariant etale neighborhood W → X of x.

This is a partial generalization of another result of Sumihiro [Sum74, Lemma. 8],[Sum75, Thm. 3.8]. He proves the existence of an open G-equivariant covering byquasi-projective subschemes when X is a normal scheme and G is connected.


2.4. Bia lynicki-Birula decompositions. In [Opr06, Prop. 5], Oprea proved theexistence of a Bia lynicki-Birula decomposition [BB73] for a smooth Deligne–Mumfordstack X with a Gm-action provided that there exists a Gm-equivariant, separated,etale atlas SpecA→ X. Therefore, Theorem 2.5 implies:

Theorem 2.7. Let X be a smooth, proper Deligne–Mumford stack over k with sep-arated diagonal, equipped with a Gm-action. Let F = tiFi be the decomposition ofthe fixed substack into connected components. Then X decomposes into disjoint, lo-cally closed, Gm-equivariant substacks Xi which are Gm-equivariant affine fibrationsover Fi.

This is proven in [Sko13, Thm. 3.5] under the additional assumptions that X istame, and the coarse moduli space of X is a scheme. See also Section 2.11 for asimilar statement due to Drinfeld for algebraic spaces with a Gm-action.

2.5. Existence of equivariant versal deformations of curves. By a curve, wemean a proper scheme over k of pure dimension one.

Theorem 2.8. Let C be an n-pointed curve. Suppose that every connected compo-nent of C is either reduced of arithmetic genus g 6= 1 or contains a marked point.Suppose that a linearly reductive group scheme H acts on C. If Aut(C) is smooth,then there exist an affine scheme W of finite type over k with an action of H fixinga point w ∈W and a miniversal deformation

C

Coo

W

Spec kwoo

of C ∼= Cw such that there exists an action of H on the total family C compatiblewith the action of H on W and Cw.

The theorem above was proven for Deligne–Mumford semistable curves in [AK16].

2.6. Good moduli spaces. In the following result, we determine the etale-localstructure of good moduli space morphisms.

Theorem 2.9. Let X be a noetherian algebraic stack over k such that X → X isa good moduli space with affine diagonal. If x ∈ X(k) is a closed point, then thereexists an affine scheme SpecA with an action of Gx and a cartesian diagram

[SpecA/Gx] //

X

π

SpecA//Gx //

X

such that SpecA//Gx → X is an etale neighborhood of π(x).

The following corollary answers negatively a question of Geraschenko–Zureick-Brown [GZB15, Qstn. 32]: does there exist an algebraic stack, with affine diagonaland good moduli space a field, that is not a quotient stack? In the equicharacteristicsetting, this result also settles a conjecture of theirs: formal GAGA holds for goodmoduli spaces with affine diagonal [GZB15, Conj. 28]. The general case will betreated in forthcoming work.

Corollary 2.10. Let X be a noetherian algebraic stack over a field k (not assumedto be algebraically closed) with affine diagonal. Suppose there exists a good modulispace π : X→ SpecR, where (R,m) is a complete local ring.

(1) Then X ∼= [SpecB/GLn]; in particular, X has the resolution property; and


(2) the natural functor

Coh(X)→ lim←−Coh(X×SpecR SpecR/mn+1


Remark 2.11. If k is algebraically closed, then in (1) above, X is in fact isomorphicto a quotient stack [SpecA/Gx] where Gx is the stabilizer of the unique closedpoint. If in addition X is smooth, then X ∼= [Nx/Gx] where Nx is the normal spaceto x (or equivalently the tangent space of X at x if Gx is smooth).

2.7. Existence of coherent completions. Let X be a noetherian algebraic stack

with affine stabilizers and Z ⊆ X be a closed substack. Denote by X[n]Z the nth

nilpotent thickening of Z ⊆ X. We say that X is coherently complete along Z if thenatural functor

Coh(X)→ lim←−n

Coh(X

[n]Z

)is an equivalence of categories. When |Z| consists of a single point x, we say that(X, x) is a complete local stack if X is coherently complete along the residual gerbeGx. See Section 3.1 for more details on coherent completion. The next result assertsthat the coherent completion always exists under very mild hypotheses.

Theorem 2.12. Let X be a quasi-separated algebraic stack, locally of finite type overk, with affine stabilizers. For any point x ∈ X(k) with linearly reductive stabilizer

Gx, there exists a complete local stack (Xx, x) and a morphism η : (Xx, x)→ (X, x)inducing isomorphisms of nth infinitesimal neighborhoods of x and x. The pair

(Xx, η) is unique up to unique 2-isomorphism.

We call Xx the coherent completion of X at x. If (W = [SpecA/Gx], w)→ (X, x)is an etale morphism as in Theorem 1.2 and π : W → W = SpecAGx is the goodmoduli space of W, then

Xx = W×W Wπ(w) = W×W Spec AGx .

That is, Xx = [SpecB/Gx] where B = A⊗AGx AGx and AGx denotes the completionat π(w) (Theorem 1.3). In particular, BG → B is of finite type and BG is thecompletion of an algebra of finite type over k.

Remark 2.13. If there exists an etale neighborhood f : W = [SpecA/Gx] → X ofx such that AGx = k, then the pair (W, f) is unique up to unique 2-isomorphism.This follows from Theorem 2.12 as W is the coherent completion of X at x.

The henselization of X at x is the stack Xhx = W×W Spec(AGx)h. This stack also

satisfies a universal property (initial among pro-etale neighborhoods of the residualgerbe at x) and will be treated in forthcoming work.

2.8. Etale-local equivalences. Before we state the next result, let us recall thatif X is an algebraic stack, locally of finite type over k, and x ∈ X(k) is a point, then

a formal miniversal deformation space of x is a formal affine scheme Def(x) together

with a formally smooth morphism Def(x)→ X which is an isomorphism on tangent

spaces. If the stabilizer group scheme Gx is smooth and linearly reductive, Def(x)inherits an action of Gx.

Theorem 2.14. Let X and Y be quasi-separated algebraic stacks, locally of finitetype over k, with affine stabilizers. Suppose x ∈ X(k) and y ∈ Y(k) are points withsmooth linearly reductive stabilizer group schemes Gx and Gy, respectively. Thenthe following are equivalent:


(1) There exist an isomorphism Gx → Gy of group schemes and an isomor-

phism Def(x) → Def(y) of formal miniversal deformation spaces which isequivariant with respect to Gx → Gy.

(2) There exists an isomorphism Xx → Yy.(3) There exist an affine scheme SpecA with an action of Gx, a point w ∈

SpecA fixed by Gx, and a diagram of etale morphisms

[SpecA/Gx]f

yy

g

%%X Y

such that f(w) = x and g(w) = y, and both f and g induce isomorphismsof stabilizer groups at w.

If additionally x ∈ |X| and y ∈ |Y| are smooth, then the conditions above areequivalent to the existence of an isomorphism Gx → Gy of group schemes and anisomorphism TX,x → TY,y of tangent spaces which is equivariant under Gx → Gy.

Remark 2.15. If the stabilizers Gx and Gy are not smooth, then the theorem aboveremains true (with the same argument) if the formal miniversal deformation spacesare replaced with flat adic presentations (Definition A.4) and the tangent spacesare replaced with normal spaces.

2.9. Characterization of when X admits a good moduli space. Using theexistence of completions, we can give an intrinsic characterization of those algebraicstacks that admit a good moduli space.

We will need one preliminary definition. We say that a geometric point y : Spec l→X is geometrically closed if the image of (y, id) : Spec l → X ×k l is a closed pointof |X×k l|.

Theorem 2.16. Let X be an algebraic stack, locally of finite type over k, with affinediagonal. Then X admits a good moduli space if and only if

(1) For every point y ∈ X(k), there exists a unique closed point in the closure

y.(2) For every closed point x ∈ X(k), the stabilizer group scheme Gx is linearly

reductive and the morphism Xx → X from the coherent completion of X atx satisfies:

(a) The morphism Xx → X is stabilizer preserving at every point; that is,

Xx → X induces an isomorphism of stabilizer groups for every point

ξ ∈ |Xx|.(b) The morphism Xx → X maps geometrically closed points to geometri-

cally closed points.

(c) The map Xx(k)→ X(k) is injective.

Remark 2.17. The quotient [P1/Gm] (where Gm acts on P1 via multiplication) doesnot satisfy (1). If X = [X/Z2] is the quotient of the non-separated affine line Xby the Z2-action which swaps the origins (and acts trivially elsewhere), then the

map Spec k[[x]] = X0 → X from the completion at the origin does not satisfy (2a).If X = [(A2 \ 0)/Gm] where Gm-acts via t · (x, y) = (x, ty) and p = (0, 1) ∈ |X|,then the map Spec k[[x]] = Xp → X does not satisfy (2b). If X = [C/Gm] whereC is the nodal cubic curve with a Gm-action and p ∈ |X| denotes the image of the

node, then [Spec(k[x, y]/xy)/Gm] = Xp → X does not satisfy (2c). (Here Gm actson coordinate axes via t · (x, y) = (tx, t−1y).) These pathological examples in factappear in many natural moduli stacks; see [AFS17, Appendix A].


Remark 2.18. Consider the non-separated affine line as a group scheme G → A1

whose generic fiber is trivial but the fiber over the origin is Z2. In this case (2a) isnot satisfied. Nevertheless, the stack quotient X = [A1/G] does have a good modulispace X = A1 but X→ X has non-separated diagonal.

Remark 2.19. When X has finite stabilizers, then conditions (1), (2b) and (2c) arealways satisfied. Condition (2a) is satisfied if and only if the inertia stack is finiteover X. In this case, the good moduli space of X coincides with the coarse space ofX, which exists by [KM97].

2.10. Algebraicity results. In this subsection, we fix a field k (not necessarilyalgebraically closed), an algebraic space S locally of finite type over k, and analgebraic stack W of finite type over S with affine diagonal over S such that W→ Sis a good moduli space. We prove the following algebraicity results.

Theorem 2.20 (Stacks of coherent sheaves). The S-stack CohW/S, whose objects

over S′ → S are finitely presented quasi-coherent sheaves on W×S S′ flat over S′,is an algebraic stack, locally of finite type over S, with affine diagonal over S.

Corollary 2.21 (Quot schemes). If F is a quasi-coherent OW-module, then theS-sheaf Quot

W/S(F), whose objects over S′ → S are quotients p∗1F → G (where

p1 : W ×S S′ → W) such that G is a finitely presented quasi-coherent OW×SS′-module flat over S′, is a separated algebraic space over S. If F is coherent, thenQuot

W/S(F) is locally of finite type.

Corollary 2.22 (Hilbert schemes). The S-sheaf HilbW/S, whose objects over S′ →S are closed substacks Z ⊆W×S S′ such that Z is flat and locally of finite presen-tation over S, is a separated algebraic space locally of finite type over S.

Theorem 2.23 (Hom stacks). Let X be a quasi-separated algebraic stack, locallyof finite type over S with affine stabilizers. If W → S is flat, then the S-stackHomS(W,X), whose objects are pairs consisting of a morphism S′ → S of alge-braic spaces and a morphism W ×S S′ → X of algebraic stacks over S, is an alge-braic stack, locally of finite type over S, with quasi-separated diagonal. If X → Shas affine (resp. quasi-affine, resp. separated) diagonal, then the same is true forHomS(W,X)→ S.

A general algebraicity theorem for Hom stacks was also considered in [HLP14,Thm. 1.6]. In the setting of Theorem 2.23 without the assumption that W→ S is agood moduli space, [HLP14, Thm. 1.6] establishes the algebraicity of HomS(W,X)under the additional hypotheses that X has affine diagonal and that W → S iscohomologically projective [HLP14, Defn. 1.12]. We note that as a consequenceof Theorem 2.9, if W → S is a good moduli space, then W → S is necessarilycohomologically projective.

We also prove the following, which we have not seen in the literature before.

Corollary 2.24 (G-equivariant Hom sheaves). Let W , X and S be quasi-separatedalgebraic spaces, locally of finite type over k. Let G be a linearly reductive affinegroup scheme acting on W and X. Let W → S and X → S be G-invariantmorphisms. Suppose that W → S is flat and a good GIT quotient. Then the S-sheaf HomG

S (W,X), whose objects over S′ → S are G-equivariant S-morphismsW ×S S′ → X, is a quasi-separated algebraic space, locally of finite type over S.

2.11. Drinfeld’s results on algebraic spaces with Gm-actions. Let Z be aquasi-separated algebraic space, locally of finite type over a field k (not assumedto be algebraically closed), with an action of Gm. Define the following sheaves on


Sch/k:

Z0 := HomGm(Spec k, Z) (the fixed locus)

Z+ := HomGm(A1, Z) (the attractor)

where Gm acts on A1 by multiplication, and define the sheaf Z on Sch/A1 by

Z := HomGm

A1 (A2, Z × A1)

where Gm acts on A2 via t · (x, y) = (tx, t−1y) and acts on A1 trivially, and themorphism A2 → A1 is defined by (x, y) 7→ xy.

Theorem 2.25. [Dri13, Prop. 1.2.2, Thm. 1.4.2 and Thm. 2.2.2] With the hypothe-

ses above, Z0, Z+ and Z are quasi-separated algebraic spaces locally of finite typeover k. Moreover, the natural morphism Z0 → Z is a closed immersion, and thenatural morphism Z+ → Z0 obtained by restricting to the origin is affine.

The algebraicity follows directly from Corollary 2.24. The final statements followfrom Theorem 2.4 above and Lemma 5.8 proved in Section 5.12.

2.12. The resolution property holds etale-locally.

Theorem 2.26. Let X be a quasi-separated algebraic stack, of finite type over aperfect (resp. arbitrary) field k, with affine stabilizers. Assume that for every closedpoint x ∈ |X|, the unit component G0

x of the stabilizer group scheme Gx is linearlyreductive. Then there exists

(1) a finite field extension k′/k;(2) a linearly reductive group scheme G over k′;(3) a k′-algebra A with an action of G; and(4) an etale (resp. quasi-finite flat) surjection p : [SpecA/G]→ X.

Moreover,

(a) If X has affine diagonal, then p can be arranged to be affine.(b) We can replace G with GLn (which is linearly reductive in characteristic

zero).

A stack of the form [SpecA/GLn] has the resolution property, that is, everycoherent sheaf is a quotient of a vector bundle [Tot04]. Although we do not knowif X has the resolution property, we conclude that X has the resolution propertyetale-locally. In [Ryd15, Def. 2.1], an algebraic stack X having the resolution prop-erty locally for a representable (resp. representable and separated) etale coveringp : W → X is said to be of global type (resp. s-global type). Thus, if X has linearlyreductive stabilizers at closed points and affine diagonal, then X is of s-global type.

Geraschenko and Satriano define toric Artin stacks in terms of stacky fans. Theyshow that a stack X is toric if and only if it is normal, has affine diagonal, has anopen dense torus T acting on the stack, has linearly reductive stabilizers, and [X/T ]is of global type [GS15, Thm. 6.1]. If X has linearly reductive stabilizers at closedpoints, then so has [X/T ]. Theorem 2.26 thus shows that the last condition issuperfluous.

2.13. Compact generation of derived categories. For results involving de-rived categories of quasi-coherent sheaves, perfect (or compact) generation of theunbounded derived category DQCoh(X) continues to be an indispensable tool at one’sdisposal [Nee96, BZFN10]. We prove:

Theorem 2.27. Let X be an algebraic stack of finite type over a field k (not assumedto be algebraically closed) with affine diagonal. If the stabilizer group Gx has linearlyreductive identity component G0

x for every closed point of X, then


(1) DQCoh(X) is compactly generated by a countable set of perfect complexes;and

(2) for every open immersion U ⊆ X, there exists a compact and perfect complexP ∈ DQCoh(X) with support precisely X \ U.

Theorem 2.27 was previously only known for stacks with finite stabilizers [HR17,Thm. A] or quotients of quasi-projective schemes by a linear action of an algebraicgroup in characteristic 0 [BZFN10, Cor. 3.22].

In positive characteristic, the theorem is almost sharp: if the reduced identitycomponent (Gx)0

red is not linearly reductive, i.e., not a torus, at some point x, thenDQCoh(X) is not compactly generated [HNR18, Thm. 1.1].

If X is an algebraic stack of finite type over k with affine stabilizers such thateither

(1) the characteristic of k is 0; or(2) every stabilizer is linearly reductive;

then X is concentrated, that is, a complex of OX-modules with quasi-coherent coho-mology is perfect if and only if it is a compact object of DQCoh(X) [HR15, Thm. C].If X admits a good moduli space π : X → X with affine diagonal, then one of thetwo conditions hold by Theorem 2.9. If X does not admit a good moduli space andis of positive characteristic, then it is not sufficient that closed points have linearlyreductive stabilizers as the following example shows.

Example 2.28. Let X = [X/Gm ×Z2] be the quotient of the non-separated affineline X by the natural Gm-action and the Z2-action that swaps the origins. ThenX has two points, one closed with stabilizer group Gm and one open point withstabilizer group Z2. Thus if k has characteristic two, then not every stabilizer groupis linearly reductive and there are non-compact perfect complexes [HR15, Thm. C].

3. Coherently complete stacks and the Tannakian formalism

3.1. Coherently complete algebraic stacks. We now prove Theorem 1.3.

Proof of Theorem 1.3. Let m ⊂ A be the maximal ideal corresponding to x. Acoherent OX-module F corresponds to a finitely generated A-module M with anaction of G. Note that since G is linearly reductive, MG is a finitely generated AG-module. We claim that the following two sequences of AG-submodules (mnM)Gand (mG)nMG of MG define the same topology, or in other words that

(3.1) MG → lim←−MG/(mnM

)Gis an isomorphism of AG-modules.

To this end, we first establish that

(3.2)⋂n≥0

(mnM

)G= 0,

which immediately informs us that (3.1) is injective. Let N =⋂n≥0 m

nM . Krull’s

intersection theorem implies that N ⊗A A/m = 0. Since AG is a local ring, SpecAhas a unique closed orbit x. Since the support of N is a closed G-invariantsubscheme of SpecA which does not contain x, it follows that N = 0.

We next establish that (3.1) is an isomorphism if AG is artinian. In this case,(mnM)G automatically satisfies the Mittag-Leffler condition (it is a sequence ofartinian AG-modules). Therefore, taking the inverse limit of the exact sequences0 → (mnM)G → MG → MG/(mnM)G → 0 and applying (3.2), yields an exactsequence

0→ 0→MG → lim←−MG/(mnM)G → 0.


Thus, we have established (3.1) when AG is artinian.To establish (3.1) in the general case, let J = (mG)A ⊆ A and observe that

(3.3) MG = lim←−MG/(mG)nMG = lim←−

(M/JnM

)G.

For each n, we know that

(3.4)(M/JnM

)G= lim←−

l

MG/((Jn + ml)M

)Gusing the artinian case proved above. Finally, combining (3.3) and (3.4) togetherwith the observation that Jn ⊆ ml for n ≥ l, we conclude that

MG = lim←−n

(M/JnM

)G= lim←−

n

lim←−l

MG/((Jn + ml)M

)G= lim←−

l

MG/(mlM

)G.

We now show that (1.1) is fully faithful. Suppose that G and F are coherentOX-modules, and let Gn and Fn denote the restrictions to X[n], respectively. Weneed to show that

Hom(G,F)→ lim←−Hom(Gn,Fn)

is bijective. Since X satisfies the resolution property, we can find locally free OX-modules E′ and E and an exact sequence

E′ → E→ G→ 0.

This induces a diagram

0 // Hom(G,F) //

Hom(E,F) //

Hom(E′,F)

0 // lim←−Hom(Gn,Fn) // lim←−Hom(En,Fn) // lim←−Hom(E′n,Fn)

with exact rows. Therefore, it suffices to assume that G is locally free. In this case,

Hom(G,F) = Hom(OX,G∨ ⊗ F) and Hom(Gn,Fn) = Hom

(OX[n] , (G∨n ⊗ Fn)

).

Therefore, we can also assume that G = OX and we need to verify that the map

Γ(X,F)→ lim←−Γ(X[n],Fn

)is an isomorphism, but this is precisely the isomorphism from (3.1), and the fullfaithfulness of (1.1) follows.

We now prove that the functor (1.1) is essentially surjective. Since X hasthe resolution property, there is a vector bundle E on X together with a surjec-tion φ0 : E → F0. We claim that φ0 lifts to a compatible system of morphismsφn : E → Fn for every n > 0. It suffices to show that for n > 0, the natural mapHom(E,Fn+1)→ Hom(E,Fn) is surjective but this is clear as Ext1

X(E,mn+1Fn+1) =0 since E is locally free and G is linearly reductive. By Nakayama’s Lemma,each φn is surjective. It follows that we obtain an induced morphism of systemsφn : En → Fn. Applying this procedure to ker(φn) (which is not necessar-ily an adic system), there is another vector bundle H and a morphism of systemsψn : Hn → En such that coker(ψn) ∼= Fn. By the full faithfulness of (1.1),

the morphism ψn arises from a unique morphism ψ : H→ E. Letting F = cokerψ,

the universal property of cokernels proves that there is an isomorphism Fn ∼= Fn;the result follows.


Remark 3.1. In this remark, we show that with the hypotheses of Theorem 1.3the coherent OX-module F extending a given system Fn ∈ lim←−Coh(X[n]) can infact be constructed explicitly. Let Γ denote the set of irreducible representationsof G with 0 ∈ Γ denoting the trivial representation. For ρ ∈ Γ, we let Vρ be thecorresponding irreducible representation. For any G-representation V , we set

V (ρ) =(V ⊗ V ∨ρ

)G ⊗ Vρ.Note that V =

⊕ρ∈Γ V

(ρ) and that V (0) = V G is the subspace of invariants.

In particular, there is a decomposition A =⊕

ρ∈ΓA(ρ). The data of a coherent

OX-module F is equivalent to a finitely generated A-module M together with aG-action, i.e., an A-module M with a decomposition M = ⊕ρ∈ΓM

(ρ), where each

M (ρ) is a direct sum of copies of the irreducible representation Vρ, such that the A-module structure on M is compatible with the decompositions of A and M . Given

a coherent OX-module F = M and a representation ρ ∈ Γ, then M (ρ) is a finitelygenerated AG-module and

M (ρ) → lim←−(M/mkM

)(ρ)is an isomorphism (which follows from (3.1)).

Conversely, given a system of Fn = Mn ∈ lim←−Coh(X[n]) where each Mn is a

finitely generated A/mn+1-module with a G-action, then the extension F = M canbe constructed explicitly by defining:

M (ρ) := lim←−M(ρ)n and M :=

⊕ρ∈Γ

M (ρ).

One can show directly that each M (ρ) is a finitely generated AG-module, M is afinitely generated A-module with a G-action, and M/mn+1M = Mn.

Remark 3.2. Theorem 1.3 also implies that every vector bundle on X is the pullbackof a G-representation under the projection X → BG. In particular, suppose thatG is a diagonalizable group scheme. Then using the notation of Remark 3.1, everyirreducible G-representation ρ ∈ Γ is one-dimensional so that a G-action on Acorresponds to a Γ-grading A =

⊕ρ∈ΓA

(ρ), and an A-module with a G-actioncorresponds to a Γ-graded A-module.

Therefore, if A =⊕

ρ∈ΓA(ρ) is a Γ-graded noetherian k-algebra with A(0) a com-

plete local k-algebra, then every finitely generated projective Γ-graded A-moduleis free. When G = Gm and AG = k, this is the well known statement (e.g.,[Eis95, Thm. 19.2]) that every finitely generated projective graded module over aNoetherian graded k-algebra A =

⊕d≥0Ad with A0 = k is free.

The theorem above motivates the following definition:

Definition 3.3. Let X be a noetherian algebraic stack with affine stabilizers andlet Z ⊆ X be a closed substack. We say that X is coherently complete along Z if thenatural functor

Coh(X)→ lim←−n

Coh(X

[n]Z


Remark 3.4. If (A,m) is complete local noetherian ring, then SpecA is coherentlycomplete along SpecA/m, and more generally if an algebraic stack X is properover SpecA, then X is coherently complete along X ×SpecA SpecA/m. See [EGA,III.5.1.4] for the case of schemes and [Ols05, Thm. 1.4], [Con05, Thm. 4.1] foralgebraic stacks. Theorem 1.3 concludes that X is coherently complete along BG.


3.2. Tannakian formalism. The following Tannaka duality theorem proved bythe second and third author is crucial in our argument.

Theorem 3.5. [HR19, Thm. 1.1] Let X be an excellent stack and Y be a noetherianalgebraic stack with affine stabilizers. Then the natural functor

Hom(X,Y)→ Homr⊗,'(Coh(Y),Coh(X)

)is an equivalence of categories, where Homr⊗,'(Coh(Y),Coh(X)) denotes the cate-gory whose objects are right exact monoidal functors Coh(Y) → Coh(X) and mor-phisms are natural isomorphisms of functors.

We will apply the following consequence of Tannakian duality:

Corollary 3.6. Let X be an excellent algebraic stack with affine stabilizers andZ ⊆ X be a closed substack. Suppose that X is coherently complete along Z. If Y isa noetherian algebraic stack with affine stabilizers, then the natural functor

Hom(X,Y)→ lim←−n

Hom(X

[n]Z ,Y


Proof. There are natural equivalences

Hom(X,Y) ' Homr⊗,'(Coh(Y),Coh(X)

)(Tannakian formalism)

' Homr⊗,'(Coh(Y), lim←−Coh

(X

[n]Z

))(coherent completeness)

' lim←−Homr⊗,'(Coh(Y),Coh

(X

[n]Z

))' lim←−Hom

(X

[n]Z ,Y

)(Tannakian formalism).

4. Proofs of Theorems 1.1 and 1.2

4.1. The normal and tangent space of an algebraic stack. Let X be a quasi-separated algebraic stack, locally of finite type over a field k, with affine stabilizers.Let x ∈ X(k) be a closed point. Denote by i : BGx → X the closed immersionof the residual gerbe of x, and by I the corresponding ideal sheaf. The normalspace to x is Nx := (I/I2)∨ = (i∗I)∨ viewed as a Gx-representation. The tangentspace TX,x to X at x is the k-vector space of equivalence classes of pairs (τ, α)consisting of morphisms τ : Spec k[ε]/ε2 → X and 2-isomorphisms α : x → τ |Spec k.The stabilizer Gx acts linearly on the tangent space TX,x by precomposition onthe 2-isomorphism. If Gx is smooth, there is an identification TX,x ∼= Nx of Gx-representations. Moreover, if X = [X/G] is a quotient stack where G is an algebraicgroup and x ∈ X(k) (with Gx not necessarily smooth), then Nx is identified withthe normal space TX,x/TG·x,x to the orbit G · x at x.

4.2. The smooth case. We now prove Theorem 1.1 even though it follows directlyfrom Theorem 1.2 coupled with Luna’s fundamental lemma [Lun73, p. 94]. We feelthat since the proof of Theorem 1.1 is more transparent and less technical thanTheorem 1.2, digesting the proof first in this case will make the proof of Theorem1.2 more accessible.

Proof of Theorem 1.1. We may replace X with an open neighborhood of x and thusassume that X is noetherian.

Define the quotient stack N = [Nx/Gx], where Nx is viewed as an affine schemevia Spec(SymN∨x ). Since Gx is linearly reductive, we claim that there are compat-ible isomorphisms X[n] ∼= N[n].

To see this, first note that we can lift X[0] = BGx to a unique morphismtn : X[n] → BGx for all n: the obstruction to a lift from tn : X[n] → BGx to


tn+1 : X[n+1] → BGx is an element of the group Ext1OBGx

(LBGx/k, In/In+1) [Ols06]

which is zero since BGx is cohomologically affine and LBGx/k is a perfect complexsupported in degrees [0, 1] since BGx → Spec k is smooth.

In particular, BGx = X[0] → X[1] has a retraction. This implies that X[1] ∼= N[1]

since both are trivial deformations by the same module. Since N→ BGk is smooth,the obstruction to lifting the morphism X[1] ∼= N[1] → N to X[n] → N vanishes asH1(BGx,Ω

∨N/BGk

⊗ In/In+1) = 0. We have induced isomorphisms X[n] ∼= N[n] by

Proposition A.7 (2).Let N→ N = Nx//Gx be the good moduli space and denote by 0 ∈ N the image

of the origin. Set N := Spec ON,0 ×N N. Since N is coherently complete (Theorem1.3), we may apply the Tannakian formalism (Corollary 3.6) to find a morphism

N→ X filling in the diagram

X[n] ∼= N[n] // ((N //

((N

X

Spec ON,0 //

N.

Let us now consider the functor F : Sch/N → Sets which assigns to a morphismS → N the set of morphisms S×N N→ X modulo 2-isomorphisms. This functor is

locally of finite presentation and we have an element of F over Spec ON,0. By Artinapproximation [Art69a, Cor. 2.2], there exist an etale morphism (U, u) → (N, 0)where U is an affine scheme and a morphism (U×NN, (u, 0))→ (X, x) agreeing with

(N, 0)→ (X, x) to first order. Since X is smooth at x, it follows that U×NN→ X isetale at (u, 0) by Proposition A.7 (2). This establishes the theorem after shrinkingU suitably; the final statement follows from Proposition 4.2.

4.3. The general case. We now prove Theorem 1.2 by a similar method to theproof in the smooth case but using Corollary A.16 in place of Artin approximation.

Proof of Theorem 1.2. We may replace X with an open neighborhood of x and thusassume that X is noetherian and that x ∈ |X| is a closed point.

Let N := [Nx/H], N→ N := Nx//H, and N := Spec ON,0×N N where 0 denotes

the image of the origin. Let η0 : BH → BGx = X[0] be the morphism induced fromthe inclusion H ⊆ Gx; this is a smooth (resp. etale) morphism. We first prove byinduction that there are compatible 2-cartesian diagrams

Hn

ηn

// Hn+1

ηn+1

X[n]

//

X[n+1],

where H0 = BH and the vertical maps are smooth (resp. etale). Indeed, givenηn : Hn → X[n], by [Ols06], the obstruction to the existence of ηn+1 is an element ofExt2

OBH(ΩBH/BGx

, η∗0(In/In+1)), but this group vanishes as H is linearly reductiveand ΩBH/BGx

is a vector bundle.Let τ0 : H0 = BH → N. Since H is linearly reductive, the deformation H0 → H1

is a trivial extension by N and hence we have an isomorphism τ1 : H1∼= N[1] (see

proof of smooth case). Using linearly reductivity of H once again and deformationtheory, we obtain compatible morphisms τn : Hn → N extending τ0 and τ1. Theseare closed immersions by Proposition A.7 (1).


The closed embeddings Hn → N factor through N so that we have a compatiblefamily of diagrams

Hn

ηn

// N

X[n] // X.

Since N is coherently complete, there exists a closed immersion H → N extending

Hn → N. Since H is also coherently complete, the Tannakian formalism yields a

morphism η : H → X extending ηn : Hn → X[n]. By Proposition A.11, H → X is

formally versal (resp. universal). Also note that H → Spec ON,0 is of finite type.We may therefore apply Corollary A.16 to obtain a stack H = [SpecA/H] together

with a morphism f : (H, w)→ (X, x) of finite type and a flat morphism ϕ : H→ H,

identifying H with the completion of H at w, such that f ϕ = η. In particular,f is smooth (resp. etale) at w. Moreover, (f ϕ)−1(X[0]) = H[0] so we have aflat morphism BH = H[0] → f−1(BGx) which equals the inclusion of the residualgerbe at w. It follows that w is an isolated point in the fiber f−1(BGx). We canthus find an open neighborhood W ⊆ H of w such that W → X is smooth (resp.etale) and W ∩ f−1(BGx) = BH. Since w is a closed point of H, we may furthershrink W so that it becomes cohomologically affine (Lemma 4.1).

The final statement follows from Proposition 4.2.

4.4. The refinement. The following trivial lemma will be frequently applied toa good moduli space morphism π : X → X. Note that any closed subset Z ⊆ X

satisfies the assumption in the lemma in this case.

Lemma 4.1. Let π : X → X be a closed morphism of topological spaces and letZ ⊆ X be a closed subset. Assume that every open neighborhood of Z containsπ−1(π(Z)). If Z ⊆ U is an open neighborhood of Z, then there exists an openneighborhood U ′ ⊆ X of π(Z) such that π−1(U ′) ⊆ U.

Proof. Take U ′ = X \ π(X \ U).

Proposition 4.2. Let f : W → X be a morphism of noetherian algebraic stackssuch that W is cohomologically affine with affine diagonal. Suppose w ∈ |W| is aclosed point such that f induces an injection of stabilizer groups at w.

(1) If there exists an affine and faithfully flat morphism of finite type X′ → X

such that X′ has quasi-finite and separated diagonal, then there exists acohomologically affine open neighborhood U ⊆ W of w such that f |U isquasi-compact, representable and separated.

(2) If X has affine diagonal, then there exists a cohomologically affine openneighborhood U ⊆W of w such that f |U is affine.

(3) If X = [X/G] where X is an algebraic space and G is an affine flat groupscheme, and f is quasi-finite, then there exists a cohomologically affine openneighborhood U ⊆W of w such that f |U is quasi-affine.

Proof. We first establish (1). By shrinking W, we may assume that ∆W/X is quasi-finite and after further shrinking, we may arrange so that W remains cohomologi-cally affine (Lemma 4.1). Let W′ = X′ ×X W and let f ′ : W′ → X′ be the inducedmorphism. Then W′ is cohomologically affine with quasi-finite and affine diago-nal. Let W′ → W ′ be the good moduli space. As ∆W ′ is a closed immersion andW′ ×W′ → W ′ × W ′ is a good moduli space, we may apply [Alp13, Prop. 6.4]to conclude that ∆W′ is finite. Since X′ has separated diagonal, it follows thatf ′ : W′ → X′ is separated. By descent, f is separated. In particular, the relative


inertia of f , i : IW/X → W, is finite. By Nakayama’s lemma, there is an opensubstack U of W, containing w, with trivial inertia relative to X. Thus U → X

is quasi-finite, representable and separated. Shrinking U appropriately, U also be-comes cohomologically affine and the claim follows.

For (2), by (1) we may assume that f is representable and separated. Since W iscohomologically affine and ∆X is affine, it follows that f is cohomologically affine.But cohomologically affine, quasi-compact, representable and separated implies thatf is affine (Serre’s Criterion [Alp13, Prop. 3.3]).

(3) follows from (1) since a representable, separated and quasi-finite morphismis quasi-affine.

Note that the condition in (1) implies that ∆X is quasi-affine. It is possible thatthe condition in (1) can be replaced by this weaker condition.

As mentioned in the Introduction, when H is diagonalizable and the diagonalof X is separated, it can also be arranged that the morphism f in Theorem 1.2 isrepresentable.

Proposition 4.3. Let S be a noetherian scheme. Let f : W→ X be a morphism oflocally noetherian algebraic stacks over S. Assume that X has separated diagonaland that W = [W/H], where W is affine over S and H is of multiplicative typeover S. If w ∈ W is fixed by H and f induces an injection of stabilizer groups atw, then there exists an H-invariant affine open U of w in W such that [U/H]→ X

is representable.

Proof. There is an exact sequence of groups over W:

1 // IW/X// IW/S

// IX/S ×X W.

Since f induces an injection of stabilizer groups at w, it follows that (IW/X)w istrivial. Also, since IX/S → X is separated, IW/X → IW/S is a closed immersion.

Let I = IW/X ×W W and pull the inclusion IW/X → IW/S back to W . SinceIW/S ×W W → H ×S W is a closed immersion, it follows that I → H ×S W isa closed immersion. Since H → S is of multiplicative type and Iw is trivial, itfollows that I → W is trivial in a neighborhood of w [SGA3II, Exp. 9, Cor. 6.5].By shrinking this open subset using Lemma 4.1, we obtain the result.

5. Proofs of Applications

We now prove the results stated in §2.

5.1. Generalization of Sumihiro’s theorem. We first establish Theorems 2.4–2.6 since they will be used to establish Theorem 2.1.

Proof of Theorem 2.4. Let Y = [X/T ] and y ∈ Y(k) be the image of x. As thesequence (2.1) splits, we can consider Tx as a subgroup of Gy. By applying Theorem1.2 to Y at y with respect to the subgroup Tx ⊆ Gy, we obtain a smooth (resp.etale) morphism f : [W/Tx] → Y, where W is an affine scheme with an action ofTx, which induces the given inclusion Tx ⊆ Gy at stabilizer groups at a preimagew ∈ [W/Tx] of y. Consider the cartesian diagram

[W/Tx]×Y X //

X

// Spec k

[W/Tx] // Y // BT

The map [W/Tx] → Y → BT induces the injection Tx → T on stabilizers groupsat w. Thus, by Proposition 4.2 (2), there is an open neighborhood U ⊆ [W/Tx] of


w such that U is cohomologically affine and U → BT is affine. The fiber productX ×Y U is thus an affine scheme SpecA and the induced map SpecA → X isT -equivariant. If u ∈ SpecA is a closed point above w and x, then the mapSpecA→ X induces an isomorphism Tx → Tx of stabilizer groups at u.

Proof of Theorem 2.5. In the exact sequence (2.1), Gx is etale and Tx is diagonal-izable. This implies that (Gy)0 is diagonalizable. Indeed, first note that we haveexact sequences:

1→ Gx ∩ (Gy)0 → (Gy)0 → (Tx)0 → 1

1→ Gx ∩ (Gy)0 → (Gy)0red → (Tx)0

red → 1

The second sequence shows that (Gy)0red is a torus (as it is connected, reduced

and surjects onto a torus with finite kernel) and, consequently, that Gx ∩ (Gy)0

is diagonalizable. It then follows that (Gy)0 is diagonalizable from the first se-quence [SGA3II, Exp. 17, Prop. 7.1.1 b)].

Theorem 1.2 produces an etale neighborhood f : ([SpecA/(Gy)0], w) → (Y, y)such that the induced morphism on stabilizers groups is (Gy)0 → Gy. ReplacingX → Y with the pull-back along f , we may thus assume that Gy is connected anddiagonalizable.

If we let Gy = D(N), Tx = D(M) and T = D(Zr), then we have a surjectivemap q : Zr → M and an injective map ϕ : M → N . The quotient N/M is torsionbut without p-torsion, where p is the characteristic of k. Since all torsion of M andN is p-torsion, we have that ϕ induces an isomorphism of torsion subgroups. Wecan thus find splittings of ϕ and q as in the diagram

Zr =Zs ⊕M/Mtor α=id⊕ϕ2 //

q=q1⊕id

Zs ⊕N/Ntor= Zr

q′=ϕ1q1⊕id

M =Mtor ⊕M/Mtor ϕ=ϕ1⊕ϕ2 // Ntor ⊕N/Ntor= N.

The map q′ corresponds to an embedding Gy → T and the map α to a reparame-terization T → T . After reparameterizing the action of T on X via α, the surjectionGy Tx becomes split. The result now follows from Theorem 2.4.

Proof of Theorem 2.6. By Theorem 1.2, there exists an etale neighborhood f : (W, w)→([X/G], x) such that W is cohomologically affine, f induces an isomorphism of sta-bilizers at w, and w is a closed point. By Proposition 4.2 (2), we can assume aftershrinking W that the composition W → [X/G] → BG is affine. It follows thatW = W×[X/G]X is affine and that W → X is an G-equivariant etale neighborhoodof x.

5.2. Generalization of Luna’s etale slice theorem.

Proof of Theorem 2.1. By applying Theorem 2.6, we can find an affine scheme X ′

with an action of G and a G-equivariant, etale morphism X ′ → X. This reducesthe theorem to the case when X is affine which was established in [Lun73, p. 97],cf. Remark 2.2.

5.3. Bia lynicki-Birula decompositions. Theorem 2.7 follows immediately fromTheorem 2.5 and [Opr06, Prop. 5].


5.4. Existence of equivariant versal deformations for curves. Theorem 2.8follows directly from Theorem 1.2 and the following lemma (because the image ofH in Aut(C) is linearly reductive and BH → BAut(C) is smooth):

Lemma 5.1. If (C, pjnj=1) is an n-pointed proper scheme of pure dimension 1over an algebraically closed field k and no connected component of Cred is a smoothunpointed irreducible curve of genus 1, then Aut(C, pj) is an affine group schemeover k.

Proof. We first handle the case when C is reduced. Let (C, pjnj=1, qjkj=1) bethe pointed normalization of (C, pj). The subgroup K ⊆ Aut(C, pj) of auto-morphisms fixing the singular locus of C has finite index, and there is an injective

homomorphism h : K → Aut(C, pj, qj). As the automorphism group of anyn-pointed smooth genus g curve with (g, n) 6= (1, 0) is affine, the hypotheses imply

Aut(C, pj, qj) is affine. It follows that K is affine and thus Aut(C, pj).In our argument for the general case, the marked points will not play a role and

will be dropped from the notation. As Cred → C can be factored by square-zeroclosed immersions, by induction, it suffices to verify the following claim: if C → D isa closed immersion of proper curves defined by an ideal sheaf I ⊆ OD such that I2 =0 and such that Aut(C) is affine, then Aut(D) is affine. Let K1 = ker(Aut(D) →Aut(C)). Since Aut(C) is affine, it suffices to prove that K1 is affine. Any elementof K1 induces naturally an automorphism of the coherent OC-module I = I/I2.Since AutOC

(I) is affine, it suffices to show that K2 = ker(K1 → AutOC(I)) is

affine. Each element α ∈ K2 defines naturally an OD-derivation

OC → I, s 7→ α(s)− s

since α is acting trivial on I and induces the identity on OD. The vector spaceDerOD

(OC , I) is finite dimensional and there is an injective group homomorphismK2 → DerOD

(OC , I), and we conclude that K2 is affine.

Remark 5.2. From the lemma above, we see that the conclusion of Theorem 2.8holds for pointed curves C such that C and every deformation of C has no connectedcomponent whose reduction is a smooth unpointed irreducible curve of genus 1.

Remark 5.3. If C → S is a family of curves such that every fiber satisfies thehypothesis of Lemma 5.1, then the automorphism group scheme Aut(C/S)→ S ofC over S need not be affine (or even quasi-affine). This even fails for families ofDeligne–Mumford semistable curves; see [AK16, §4.1].

5.5. Good moduli spaces.

Proof of Theorem 2.9. By Theorem A.1, X→ X is of finite type. We may assumethat X = SpecR, where R is a noetherian k-algebra. By noetherian approximationalong k → R, there is a finite type k-algebra R0 and an algebraic stack X0 of finitetype over SpecR0 with affine diagonal. We may also arrange that the image x0 ofx in X0 is closed with linearly reductive stabilizer Gx. We now apply Theorem 1.2to find a pointed affine etale k-morphism f0 : ([SpecA0/Gx], w0) → (X0, x0) thatinduces an isomorphism of stabilizers at w0. Pulling this back along SpecR →SpecR0, we obtain an affine etale morphism f : [SpecA/Gx] → X inducing anisomorphism of stabilizers at all points lying over the preimage of w0. The resultnow follows from a generalization of Luna’s fundamental lemma [Alp10, Thm. 6.10].

Proof of Corollary 2.10. By [GZB15, Thm. 1], we have (1) ⇒ (2); thus, it sufficesto prove (1). If R/m = k and k is algebraically closed, then the result follows from


Theorem 2.9 since GLn/Gx is affine for any embedding Gx → GLn. In this case(1) holds even if R is not complete but merely henselian.

If R/m = k and k is not algebraically closed, then we proceed as follows. Let k bean algebraic closure of k. By [EGA, 0III.10.3.1.3], R = R⊗k k = lim−→k⊆k′⊆k R⊗k k

′

is local, noetherian, m = mR is the maximal ideal, R/m ∼= k, and the induced mapR/m→ R/m coincides with k → k. Since each R⊗k k′ is henselian, R is henselian.Let X = X ⊗R R. By the case considered above, X has the resolution property.Having the resolution property descends to some Xk′ = X⊗k k′, where k ⊆ k′ ⊆ kis a finite extension. Since Xk′ → X is finite and faithfully flat, X has the resolutionproperty [Gro17, Prop. 5.3(vii)].

In general, let K = R/m. Since R is a complete k-algebra, it admits a coefficientfield; thus, it is also a K-algebra. We are now free to replace k with K and theresult follows.

5.6. Existence of coherent completions. Theorem 1.2 gives an etale morphism(W = [SpecA/Gx], w) → (X, x). If we let π : W → W = SpecAGx , then Theorem

2.12 follows by taking Xx = W ×W Spec OW,π(w). Indeed, this stack is coherentlycomplete by Theorem 1.3 and the uniqueness follows by the Tannakian formalism(Corollary 3.6).

5.7. Etale-local equivalences.

Proof of Theorem 2.14. The implications (3) =⇒ (2) =⇒ (1) are immediate. We

also have (1) =⇒ (2) as X[n] = [Def(x)[n]/Gx] and Y[n] = [Def(y)[n]/Gy]. We now

show that (2) =⇒ (3). We are given an isomorphism α : Xx∼→ Yy. Let (W =

[SpecA/Gx], w) → (X, x) be an etale neighborhood as in Theorem 1.2. Let W =SpecAGx denote the good moduli space of W and let w0 be the image of w. Then

Xx = W×W Spec OW,w0. The functor F : (T →W ) 7→ Hom(W×W T,Y) is locally of

finite presentation. Artin approximation applied to F and α ∈ F (Spec OW,w0) thus

gives an etale morphism (W ′, w′)→ (W,w) and a morphism ϕ : W′ := W×WW ′ →Y such that ϕ|W′[1] : W′[1] → Y[1] is an isomorphism. Since W′w′ ∼= Xx ∼= Yy, it

follows that ϕ induces an isomorphism W′ → Y by Proposition A.9. After replacingW ′ with an open neighborhood we thus obtain an etale morphism (W′, w′)→ (Y, y).The final statement is clear from Theorem 1.1.

5.8. Characterization of when X admits a good moduli space.

Proof of Theorem 2.16. The necessity of the conditions follow from Theorem 2.9.For the sufficiency, by [AFS17, Proposition 3.1], it is enough to verify:

(I) For every closed point x ∈ |X|, there exists an affine etale morphism

f :([SpecA/Gx], w

)→ (X, x)

such that for each closed point w′ ∈ [SpecA/Gx],(a) f is stabilizer preserving at w′ (i.e., f induces an isomorphism of sta-

bilizer groups at w′); and(b) f(w′) is closed.

(II) For any point y ∈ X(k), the closed substack y admits a good modulispace.

We first verify condition (I). Let x ∈ X(k) be a closed point. By Theorem 1.2, thereexist a quotient stack W = [SpecA/Gx] with a closed point w ∈ |W| and an affineetale morphism f : (W, w) → (X, x) such that f is stabilizer preserving at w. As


the coherent completion of W at w is identified with Xx, we have a 2-commutativediagram

Xx

W

f // X.

(5.1)

The subset Qa ⊆ |W| consisting of points ξ ∈ |W| such that f is stabilizer preserving

at ξ is constructible. Since Qa contains every point in the image of Xx → W

by hypothesis (2a), it follows that Qa contains a neighborhood of w. Thus afterreplacing W with an open saturated neighborhood containing w (Lemma 4.1), wemay assume that f : W→ X satisfies condition (Ia).

Let π : W → W be the good moduli space of W and consider the morphismg = (f, π) = W → X ×W . For a point ξ ∈ |W |, let ξ0 ∈ |W| denote the uniquepoint that is closed in the fiber Wξ. Let Qb ⊆ |W | be the locus of points ξ ∈ |W |such that g(ξ0) is closed in |(X×W )ξ| = |Xκ(ξ)|. This locus is constructible. Indeed,

the subset W0 = ξ0 : ξ ∈ |W | ⊆ |W| is easily seen to be constructible; henceso is g(W0) by Chevalley’s theorem. The locus Qb equals the set of points ξ ∈ |W |such that g(W0)ξ is closed which is constructible by [EGA, IV.9.5.4]. The locus Qbcontains SpecOW,π(w) by hypothesis (2b) (recall that Xx = W ×W Spec OW,π(w)).Therefore, after replacing W with an open saturated neighborhood of w, we mayassume that f : W→ X satisfies condition (Ib).

For condition (II), we may replace X by y. By (1), there is a unique closed

point x ∈ y and we can find a commutative diagram as in (5.1) for x. By (2b)we can, since f is etale, also assume that W has a unique closed point. This

implies that Γ(W,OW) = k and Xx = W. By hypothesis (2c), f : W → X is anetale monomorphism which is also surjective by hypothesis (1). We conclude thatf : W→ X is an isomorphism establishing condition (II).

5.9. The resolution property holds etale-locally.

Proof of Theorem 2.26. First assume that k is algebraically closed. Since X isquasi-compact, Theorem 1.2 gives an etale surjective morphism q : [U1/G1]q · · · q[Un/Gn]→ X where Gi is a linearly reductive group scheme over k acting on affinescheme Ui. If we let G = G1×G2× · · · ×Gn and let U be the disjoint union of theUi ×G/Gi, we obtain an etale surjective morphism p : [U/G] → X. If X has affinediagonal, then we can assume that q, and hence p, are affine.

For general k, write the algebraic closure k as a union of its finite subextensionsk′/k. A standard limit argument gives a solution over some k′.

Since G is reductive, the quotient GLn/G is affine for any embedding G → GLn.Note that [U/G] = [U ×GGLn/GLn] and U ×GGLn is affine since U ×GGLn → Uis a GLn/G-fibration.

5.10. Compact generation of derived categories. Theorem 2.27 follows im-mediately from Theorem 2.26 together with [HR17, Thm. B] (characteristic 0) or[HR15, Thm. D] (positive characteristic).

5.11. Algebraicity results. These will be established using Artin’s criterion, asformulated in [Hal17, Thm. A]. Consequently, we will need a preparatory result oncoherence (in the sense of Auslander [Aus66]) of the relevant deformation and ob-struction functors, which will also help with the separation conditions. Throughoutthis subsection, we assume the following:

• k is a field (not necessarily algebraically closed);


• S is an algebraic space, locally of finite type over k; and• W is an algebraic stack of finite type over k with affine diagonal that admits

a good moduli space W→ S.

The following proposition is a variant of [Hal14, Thm. C] and [HR17, Thm. D].

Proposition 5.4. Assume S is an affine scheme. If F• ∈ DQCoh(W) and G• ∈DbCoh(W), then the functor

HomOW

(F•,G• ⊗L

OWLπ∗QCoh(−)

): QCoh(S)→ QCoh(S)

is coherent.

Proof. By Theorem 2.27, DQCoh(W) is compactly generated. Also, the restrictionof Rf∗,QCoh : DQCoh(W) → DQCoh(S) to D+

Coh(W) factors through D+Coh(S) [Alp13,

Thm. 4.16(x)]. By [HR17, Cor. 4.19], the result follows.

The following corollary is a variant of [Hal14, Thm. D], whose proof is identical.

Corollary 5.5. Let F be a quasi-coherent OW-module and let G be a coherent OW-module. If G is flat over S, then the S-presheaf HomOW/S(F,G) whose objects over

S′τ−→ S are homomorphisms τ∗WF → τ∗WG of OW×SS′-modules (where τW : W ×S

S′ →W is the projection) is representable by an affine S-scheme.

Proof. Argue exactly as in the proof of [Hal14, Thm. D], but using Proposition5.4 in place of [Hal14, Thm. C] to deduce that automorphisms, deformations andobstructions are coherent.

Proof of Theorem 2.20. This only requires small modifications to the proof of [Hal17,Thm. 8.1]: the formal GAGA statement of Corollary 2.10 implies that formally ver-sal deformations are effective and Proposition 5.4 implies that the automorphism,deformation and obstruction functors are coherent. Therefore, Artin’s criterion (asformulated in [Hal17, Thm. A]) is satisfied and the result follows. Corollary 5.5implies the diagonal is affine.

Corollaries 2.21 and 2.22 follow immediately from Theorem 2.20. Indeed, the nat-ural functor Quot

W/S(F)→ CohW/S is quasi-affine by Corollary 5.5 and Nakayama’s

Lemma (see [Lie06, Lem. 2.6] for details).

Proof of Theorem 2.23. This only requires small modifications to the proof of [HR19,Thm. 1.2], which uses Artin’s criterion as formulated in [Hal17, Thm. A]. Indeed,using Corollary 2.10 in place of [Ols05, Thm. 1.4] (effectivity), Proposition 5.4 inplace of [Hal14, Thm. C] (coherence) and Corollary 5.5 in place of [Hal14, Thm. D](conditions on the diagonal), the result follows.

Proof of Corollary 2.24. A G-equivariant morphism W → X is equivalent to amorphism of stacks [W/G]→ [X/G] over BG. This gives the cartesian diagram

HomGS (W,X) //

HomS

([W/G], [X/G]

)

S // HomS

([W/G], BG

)

where the bottom map is given by the structure map [W/G] → BG and the rightmap is given by postcomposition with the structure map [X/G] → BG. By The-orem 2.23, the stacks HomS

([W/G], [X/G]

)and HomS

([W/G], BG

)are algebraic,

locally of finite type over S and have quasi-affine diagonals. In particular, the


bottom map is quasi-affine. It follows that HomGS (W,X) is a quasi-separated al-

gebraic stack locally of finite type over S. But HomGS (W,X) has no non-trivial

automorphisms, hence is an algebraic space.

5.12. Drinfeld’s results on algebraic spaces with Gm-actions. We begin thissubsection with the following coherent completeness lemma.

Lemma 5.6. If S is a noetherian affine scheme, then [A1S/Gm] is coherently com-

plete along [S/Gm].

Proof. Let A = Γ(S,OS); then A1S = SpecA[t] and V (t) = [S/Gm]. If F ∈

Coh([A1S/Gm]), then we claim that there exists an integer n 0 such that the

natural surjection Γ(F) → Γ(F/tnF) is bijective. Now every coherent sheaf on[A1S/Gm] is a quotient of a finite direct sum of coherent sheaves of the form p∗El,

where El is the weight l representation of Gm and p : [A1S/Gm] → [S/Gm] is the

natural map. It is enough to prove that Γ(p∗El) → Γ(p∗El/tnp∗El) is bijective,

or equivalently, that Γ((tn) ⊗ p∗El) = 0. But (tn) = p∗En and Γ(p∗En+l) = 0 ifn+ l > 0, hence for all n 0. We conclude that Γ(F)→ lim←−n Γ(F/tnF) is bijective.

What remains can be proven analogously to Theorem 1.3.

Proposition 5.7. Let W be an excellent algebraic space over a field k and G bean algebraic group acting on W . Let Z ⊆ W be a G-invariant closed subspace.Suppose that [W/G] is coherently complete along [Z/G]. Let X be a noetherianalgebraic space over k with an action of G. Then the natural map

HomG(W,X)→ lim←−n

HomG(W

[n]Z , X

)is bijective.

Proof. As in the proof of Corollary 2.24, we have a cartesian diagram

HomG(W,X) //

Hom([W/G], [X/G]

)

Spec k // Hom([W/G], BG

)and a similar cartesian diagram for W replaced with W

[n]Z for any n which gives

the cartesian diagram

lim←−n HomG(W

[n]Z , X

)//

lim←−n Hom([W

[n]Z /G], [X/G]

)

Spec k // lim←−n Hom([W

[n]Z /G], BG

)Since [W/G] is coherently complete along [Z/G], it follows by Tannaka duality thatthe natural maps from the first square to the second square are isomorphisms.

Lemma 5.8. Let f : U → Z be a Gm-equivariant etale morphism of quasi-separatedalgebraic spaces of finite type over a field k. Then U0 = Z0 ×Z U and U+ =Z+ ×Z0 U0.

Proof. The inclusion of stabilizer group schemes Stab(U) → Stab(Z) ×Z U is anopen immersion since f is etale. The first statement follows since any open subgroup


of Gm is Gm. For the second statement, we need to show that there exists a uniqueGm-equivariant morphism filling in the Gm-equivariant diagram

Spec k × S //

U

f

A1 × S //

::

Z

(5.2)

where S is an affine scheme of finite type over k, and the vertical left arrow is theinclusion of the origin. For each n ≥ 1, the formal lifting property of etalenessyields a unique Gm-equivariant map Spec k[x]/xn × S → U such that

Spec k × S //

U

f

Spec(k[x]/xn)× S //

88

Z

commutes. By Lemma 5.6 and Proposition 5.7, there exists a unique Gm-equivariantmorphism A1 × S → U such that (5.2) commutes.

Proof of Theorem 2.25. The algebraicity of Z0, Z+ and Z follows directly fromCorollary 2.24. The final statements may be verified after passing to an algebraicclosure of k. Our generalization of Sumihiro’s theorem (Theorem 2.4) and Lemma5.8 now further reduce these to the case when Z is an affine scheme, which can beestablished directly; see [Dri13, §1.3.4].

Appendix A. Equivariant Artin algebraization

In this appendix, we give an equivariant generalization of Artin’s algebraizationtheorem [Art69b, Thm. 1.6]. We follow the approach of [CJ02] relying on Popescu’stheorem that establishes general Neron desingularization: any regular homomor-phism of noetherian rings is a filtered colimit of smooth homomorphisms.

The main results of this appendix (Theorems A.14 and A.15) are formulated ingreater generality than necessary to prove Theorem 1.2. We feel that these resultsare of independent interest and will have further applications. In particular, in asubsequent article we will apply the results of this appendix to prove a relativeversion of Theorem 1.2.

A.1. Good moduli space morphisms are of finite type. Let G be a group act-ing on a noetherian ring A. Goto–Yamagishi [GY83] and Gabber [ILO14, Exp. IV,Prop. 2.2.3] have proven that A is finitely generated over AG when G is either diag-onalizable (Goto–Yamagishi) or finite and tame (Gabber). Equivalently, the goodmoduli space morphism X = [SpecA/G]→ SpecAG is of finite type. The followingtheorem generalizes this result to every noetherian stack with a good moduli space.

Theorem A.1. Let X be a noetherian algebraic stack. If π : X → X is a goodmoduli space with affine diagonal, then π is of finite type.

Proof. We may assume that X = SpecA is affine. Let p : U = SpecB → X be an

affine presentation. Then π∗(p∗OU ) = B. We need to show that B is a finitelygenerated A-algebra. This follows from the following lemma.

Lemma A.2. If X is a noetherian algebraic stack and π : X→ X is a good modulispace, then π∗ preserves finitely generated algebras.

Proof. Let A be a finitely generated OX-algebra. Write A = lim−→λFλ as a union of its

finitely generated OX-submodules. Then A is generated as an OX-algebra by Fλ forsufficiently large λ; that is, we have a surjection Sym(Fλ)→ A. Since π∗ is exact, it


is enough to prove that π∗ Sym(Fλ) is finitely generated. But C := Γ(X,Sym(Fλ)) isa Z-graded ring which is noetherian by [Alp13, Thm. 4.16(x)] since Spec

X(Sym(Fλ))

is noetherian and Spec(C) is its good moduli space. It is well-known that C is thenfinitely generated over C0 = Γ(X,OX) = A.

A.2. Artinian stacks.

Definition A.3. We say that an algebraic stack X is artinian if it is noetherianand |X| is discrete. We say that a quasi-compact and quasi-separated algebraicstack X is local if there exists a unique closed point x ∈ |X|.

Let X be a noetherian algebraic stack and let x ∈ |X| be a closed point withmaximal ideal mx ⊂ OX. The nth infinitesimal neighborhood of x is the closedalgebraic stack X[n] → X defined by mn+1

x . Note that X[n] is artinian and thatX[0] = Gx is the residual gerbe. A local artinian stack X is a local artinian schemeif and only if X[0] is the spectrum of a field.

Definition A.4. Let X and Y be algebraic stacks, and let x ∈ |X| and y ∈ |Y| beclosed points. We say that a morphism f : (X, x)→ (Y, y) is adic if f−1(Y[0]) = X[0].

Note that f is adic precisely when f∗my → mx is surjective. When f is adic, we

thus have that f−1(Y[n]) = X[n] for all n ≥ 0. Every closed immersion is adic.

Proposition A.5. Let X be a quasi-separated algebraic stack and let x ∈ |X| bea closed point. Then there exists an adic flat presentation, that is, there exists anadic flat morphism of finite presentation p : (SpecA, u) → (X, x). If the stabilizergroup of x is smooth, then there exists an adic smooth presentation.

Proof. The question is local on X so we can assume that X is quasi-compact. Startwith any smooth presentation q : V = SpecA → X. The fiber Vx = q−1(Gx) =SpecA/I is smooth over the residue field κ(x) of the residual gerbe. Pick a closedpoint v ∈ Vx such that κ(v)/κ(x) is separable. After replacing V with an openneighborhood of v, we may pick a regular sequence f1, f2, . . . , fn ∈ A/I thatgenerates mv. Lift this to a sequence f1, f2, . . . , fn ∈ A and let Z → V be theclosed subscheme defined by this sequence. The sequence is transversely regularover X in a neighborhood W ⊆ V of v. In particular, U = W ∩ Z → V → X

is flat. By construction Ux = Zx = Specκ(v) so (U, v) → (X,x) is an adic flatpresentation. Moreover, Specκ(v) → Gx → κ(x) is etale so if the stabilizer groupis smooth, then Ux → Gx is smooth and U → X is smooth at u.

Corollary A.6. Let X be a noetherian algebraic stack. The following statementsare equivalent.

(1) There exists an artinian ring A and a flat presentation p : SpecA → X

which is adic at every point of X.(2) There exists an artinian ring A and a flat presentation p : SpecA→ X.(3) X is artinian.

Proof. The implications (1) =⇒ (2) =⇒ (3) are trivial. The implication (3) =⇒ (1)follows from the proposition.

If X is an algebraic stack, I ⊆ OX is a sheaf of ideals and F is a quasi-coherentsheaf, we set GrI(F) := ⊕n≥0I

nF/In+1F, which is a quasi-coherent sheaf of gradedmodules on the closed substack defined by I.

Proposition A.7. Let f : (X, x)→ (Y, y) be a morphism of local artinian stacks.

(1) f is a closed immersion if and only if f |X[1] : X[1] → Y is a closed immersion.


(2) f is an isomorphism if and only if f is a closed immersion and there ex-ists an isomorphism ϕ : Grmx

(OX) → (f [0])∗Grmy(OY ) of graded OX[0]-

modules.

Proof. The conditions are clearly necessary. For the sufficiency, pick an adic flatpresentation p : SpecA→ Y. After pulling back f along p, we may assume that Y =SpecA is a local artinian scheme. If f |X[0] is a closed immersion, then X = SpecBis also a local artinian scheme. If in addition f |X[1] is a closed immersion, thenmA → mB/m

2B is surjective; hence so is mAB → mB by Nakayama’s Lemma. We

conclude that f is adic and that A→ B is surjective (Nakayama’s lemma again).Assume that in addition we have an isomorphism ϕ : GrmA

A ∼= GrmBB of

graded k-vector spaces where k = A/mA = B/mB . Then dimk mnA/m

n+1A =

dimk mnB/m

n+1B . It follows that the surjections mnA/m

n+1A → mnB/m

n+1B induced

by f are isomorphisms. It follows that f is injective.

Recall from Section 2.7 that a complete local stack is a noetherian local stack(X, x) such that X is coherently complete along the residual gerbe Gx.

Lemma A.8. Let f : (X, x) → (Y, y) be a morphism of complete local stacks thatinduces an isomorphism f [0] : X[0] → Y[0].

(1) f is a closed immersion if and only if f |X[n] : X[n] → Y is a closed immersionfor all n ≥ 0.

(2) f is an isomorphism if and only if f |Y[n] : X[n] → Y[n] is an isomorphismfor all n ≥ 0.

Proof. In the first statement, we have a compatible system of closed immersionsX[n] → Y[n]. Since Y is coherently complete, we obtain a unique closed substackZ → Y such that X[n] = Z[n] for all n. Since X and Y are coherently complete, itfollows by Tannaka duality (Theorem 3.5) that we have an isomorphism X → Z.The second statement follows directly from Tannaka duality since X and Y arecoherently complete.

Combining the lemma and Proposition A.7 we obtain.

Proposition A.9. Let f : (X, x)→ (Y, y) be a morphism of complete local stacks.

(1) f is a closed immersion if and only if f |X[1] : X[1] → Y is a closed immersion.(2) f is an isomorphism if and only if f is a closed immersion and there ex-

ists an isomorphism ϕ : Grmx(OX) → (f [0])∗Grmy (OY ) of graded OX[0]-modules.

A.3. Formal versality.

Definition A.10. Let W be a noetherian algebraic stack, let w ∈ |W| be a closedpoint and let W[n] denote the nth infinitesimal neighborhood of w. Let X be acategory fibered in groupoids and let η : W → X be a morphism. We say that ηis formally versal (resp. formally universal) at w if the following lifting conditionholds. Given a 2-commutative diagram of solid arrows

W[0] ι // Z

f // _

g

W

η

Z′ //

f ′>>

X

where Z and Z′ are local artinian stacks and ι and g are closed immersions, thereexists a morphism (resp. a unique morphism) f ′ and 2-isomorphisms such that thewhole diagram is 2-commutative.


Proposition A.11. Let η : (W, w)→ (X, x) be a morphism of noetherian algebraicstacks. Assume that w and x are closed points. Suppose that η[0] : W[0] → X[0] isrepresentable.

(1) If W[n] → X[n] is etale for every n, then η is formally universal at w.(2) If W[n] → X[n] is smooth for every n and the stabilizer Gw is linearly

reductive, then η is formally versal at w.

Proof. The first part is clear from descent. For the second part, we begin withthe following observation: if (Z, z) is a local artinian stack and h : (Z, z) → (Q, q)is a morphism of algebraic stacks, where q is a closed point, then there exists ann such that h factors through Q[n]. Now, if we are given a lifting problem, thenthe previous observation shows that we may assume that Z′ and Z factor throughsome W[n] → X[n], which is representable. The obstruction to the existence of alift belongs to the group Ext1

Z(f∗ΩW[n]/X[n] , I), where I is the square zero idealdefining the closed immersion g. But Z is cohomologically affine and ΩW[n]/X[n] isa vector bundle, so the Ext-group vanishes. The result follows.

A.4. Refined Artin–Rees for algebraic stacks. The results in this section isa generalization of [CJ02, §3] (also see [Stacks, Tag 07VD]) from rings to algebraicstacks.

Definition A.12. Let X be a noetherian algebraic stack and let Z → X be aclosed substack defined by the ideal I ⊆ OX. Let ϕ : E → F be a homomorphismof coherent sheaves on X. Let c ≥ 0 be an integer. We say that (AR)c holds for ϕalong Z if

ϕ(E) ∩ InF ⊆ ϕ(In−cE), ∀n ≥ c.

When X is a scheme, (AR)c holds for all sufficiently large c by the Artin–Reeslemma. If π : U → X is a flat presentation, then (AR)c holds for ϕ along Z if andonly if (AR)c holds for π∗ϕ : π∗E → π∗F along π−1(Z). In particular (AR)c holdsfor ϕ along Z for all sufficiently large c. If f : E′ E is a surjective homomorphism,then (AR)c for ϕ holds if and only if (AR)c for ϕ f holds.

In the following section, we will only use the case when |Z| is a closed point.

Theorem A.13. Let E2α−→ E1

β−→ E0 and E′2α′

−→ E′1β′

−→ E′0 be two complexes ofcoherent sheaves on a noetherian algebraic stack X. Let Z → X be a closed substackdefined by the ideal I ⊆ OX. Let c be a positive integer. Assume that

(1) E0,E′0,E1,E

′1 are vector bundles,

(2) the sequences are isomorphic after tensoring with OX/Ic,

(3) the first sequence is exact, and(4) (AR)c holds for α and β along Z.

Then

(a) the second sequence is exact in a neighborhood of Z;(b) (AR)c holds for β′ along Z; and(c) given an isomorphism ϕ : E0 → E′0, there exists a unique isomorphism ψ of

GrI(OX)-modules in the diagram

GrI(E0)Gr(γ) // //

Gr(ϕ)∼=

GrI(cokerβ)

ψ∼=

GrI(E′0)

Gr(γ′) // // GrI(cokerβ′)

where γ : E0 → cokerβ and γ′ : E′0 → cokerβ′ denote the induced maps.

http://stacks.math.columbia.edu/tag/07VD


Proof. Note that there exists an isomorphism ψ if and only if ker Gr(γ) = ker Gr(γ′).All three statements can thus be checked after pulling back to a presentation U → X.We may also localize and assume that X = U = SpecA where A is a local ring.Then all vector bundles are free and we may choose isomorphisms Ei ∼= E′i fori = 0, 1 such that β = β′ modulo Ic. We can also choose a surjection ε′ : OnU E′2and a lift ε : OnU E2 modulo Ic, so that α ε = α′ ε′ modulo Ic. Thus, we mayassume that Ei = E′i for i = 0, 1, 2 are free. The result then follows from [CJ02,Lem. 3.1 and Thm. 3.2] or [Stacks, Tags 07VE and 07VF].

A.5. Equivariant algebraization.

Theorem A.14. Let S be an excellent scheme and let T be a noetherian algebraicspace over S. Let Z be an algebraic stack of finite presentation over T and letz ∈ |Z| be a closed point such that Gz → S is of finite type. Let t ∈ T be theimage of z. Let X1, . . . ,Xn be categories fibered in groupoids over S, locally of finitepresentation. Let η : Z → X = X1 ×S · · · ×S Xn be a morphism. Fix an integerN ≥ 0. Then there exists

(1) an affine scheme S′ of finite type over S and a point s′ ∈ S′ mapping tothe same point in S as t ∈ T ;

(2) an algebraic stack W→ S′ of finite type;(3) a closed point w ∈ |W| over s′;(4) a morphism ξ : W→ X;(5) an isomorphism Z×T T [N ] ∼= W×S′ S′[N ] over X; in particular, there is an

isomorphism Z[N ] ∼= W[N ] over X; and(6) an isomorphism Grmz

OZ∼= Grmw

OW of graded algebras over Z[0] ∼= W[0].

Moreover, if Xi is a quasi-compact algebraic stack and ηi : Z → Xi is affine forsome i, then it can be arranged so that ξi : W→ Xi is affine.

Proof. We may assume that S = SpecA is affine. Let t ∈ T be the image of z.

By replacing T with the completion T = Spec OT,t and Z with Z ×T T , we may

assume that T = T = SpecB where B is a complete local ring. By standard limitmethods, we have an affine scheme S0 = SpecB0 and an algebraic stack Z0 → S0

of finite presentation and a commutative diagram

Z //

))Z0

//

X

T // S0

//

S

If Xi is algebraic and quasi-compact and Z → Xi is affine for some i, we may alsoarrange so that Z0 → Xi is affine [Ryd15, Thm. C].

Since Gz → S is of finite type, so is Specκ(t) → S. We may thus choose a

factorization T → S1 = AnS0→ S0, such that T → S1 = Spec OS1,s1 is a closed

immersion; here s1 ∈ S1 denotes the image of t ∈ T . Let Z1 = Z0 ×S0S1 and

Z1 = Z1 ×S1S1. Consider the functor F : Sch/S1 → Sets where F (U → S1) is the

set of isomorphism classes of complexes

E2α−→ E1

β−→ OZ1×S1U

of finitely presented quasi-coherent OZ1×S1U -modules with E1 locally free. By stan-

dard limit arguments, F is locally of finite presentation.

We have an element (α, β) ∈ F (S1) such that im(β) defines Z → Z1. Indeed,choose a resolution

O⊕rS1,s1

β−−−→ OS1,s1 B.

http://stacks.math.columbia.edu/tag/07VE

http://stacks.math.columbia.edu/tag/07VF


After pulling back β to Z1, we obtain a resolution

ker(β)α

−−→ O⊕rZ1

β−−→ OZ1 OZ.

After increasing N , we may assume that (AR)N holds for α and β at Z[0]1 .

We now apply Neron–Popescu approximation [Pop86, Thm. 1.3] to the regular

morphism S1 → S1 and obtain an etale neighborhood (S′, s′) → (S1, s1) and an

element (α′, β′) ∈ F (S′) such that (α, β) = (α′, β′) in F (S[N ]1 ). We let W →

Z1×S1S′ be the closed substack defined by im(β′). Then Z×T T [N ] and W×S′ S′[N ]

are equal as closed substacks of Z1×S1S

[n]1 and (1)–(5) follows. Finally (6) follows

from Theorem A.13.

Theorem A.15. Let S, T , Z, η, N , W and ξ be as in Theorem A.14. If η1 : Z→ X1

is formally versal, then there are compatible isomorphisms ϕn : W[n] ∼→ Z[n] overX1 for all n ≥ 0. For n ≤ N , the isomorphism ϕn is also compatible with η and ξ.

Proof. We can assume that N ≥ 1. By Theorem A.14, we have an isomorphismϕN : W[N ] → Z[N ] over X. By formal versality and induction on n ≥ N , we can ex-tend ϕN to compatible morphisms ϕ′n : W[n] → Z over X1. Indeed, formal versalityallows us to find a dotted arrow such the diagram

W[n]ϕ′

n //

Z

η

W[n+1]

ξ1|W[n+1]

//

ϕ′n+1

88

X1

is 2-commutative. By Proposition A.7, ϕ′n induces an isomorphism ϕn : W[n] →Z[n].

We now formulate the theorem above in a manner which is transparently an equi-variant analogue of Artin algebraization [Art69b, Thm. 1.6]. It is this formulationthat is directly applied to prove Theorem 1.2.

Corollary A.16. Let H be a linearly reductive affine group scheme over an alge-braically closed field k. Let X be a category fibered in groupoids over k that is locallyof finite presentation. Let Z = [SpecA/H] be a noetherian algebraic stack over k.Suppose that AH is a complete local k-algebra. Let η : Z → X be a morphism thatis formally versal at z ∈ |Z|. Let N ≥ 0. Then there exists

(1) an algebraic stack W = [SpecB/H] of finite type over k;(2) a closed point w ∈ |W|;(3) a morphism ξ : W→ X; and

(4) an isomorphism ϕ : W → Z, where W is the coherent completion of W at

w (i.e., W = W ×W Spec OW,w0where W = SpecBH and w0 ∈ W is the

image of w under W→W )

such that

(a) for all n, the induced isomorphisms ϕn : W[n] → Z[n] are compatible with ηand ξ; and

(b) ϕN : W[N ] → Z[N ] is an isomorphism over BH.

Moreover, if X is a noetherian algebraic stack with affine stabilizers, then ϕ iscompatible with η and ξ.

Proof. By Theorem A.1, Z→ SpecAH is of finite type. If we apply Theorem A.15with S = Spec k, T = SpecAH , X1 = X and X2 = BH, then we immediatelyobtain (1)–(3) together with isomorphisms ϕn : W[n] → Z[n] that are compatible


over X for all n and compatible over BH for all n ≤ N . Since Z and W arecoherently complete, by the Tannakian formalism the isomorphisms ϕn yields an

isomorphism ϕ : W → Z (see Corollary 3.6). If X is algebraic and noetherian with

affine stabilizers and π : W → W denotes the completion map, then ξ π = η ϕby the Tannakian formalism (fully faithfulness in Corollary 3.6).

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